Homotopy negligible subsets of bundles

C ^OMPOSITIO M ^ATHEMATICA

R ^AYMOND Y. T. W ^ONG

Homotopy negligible subsets of bundles

Compositio Mathematica, tome 24, n

^o

1 (1972), p. 119-128

<http://www.numdam.org/item?id=CM_1972__24_1_119_0>

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119

HOMOTOPY NEGLIGIBLE SUBSETS OF BUNDLES by

Raymond

^{Y. T.}

Wong

COMPOSITIO MATHEMATICA, ^Vol. 24, ^Fasc. 1, 1972, pag. 119-128 Wolters-Noordhoff Publishing

Printed in the Netherlands

1. Introduction

Recently

^{Eells and}

Kuiper [2] have

^shown that for any

ANR, locally homotopy negligible

^{closed subsets are}

homotopy negligible.

^It is conceiv- able that such results will be

applicable

^to

manifolds, particularly

^of

infinite-dimension. The purpose of this paper is ^to

derive,

^{in the}

setting

of

locally

trivial bundle _,a condition that a closed subset of the total space will be

locally homotopy negligible.

^A

typical

^{result is}

(for

^{a more}

precise

statement, ^see Theorem

3)

^{that for a}

^locally

trivial bundle

03BE :

^{X ~ B} with base

space B

^{and fibre F}

being

manifolds modeled

on a closed convex subset of some metric

locally

^convex

topological

vector space

(MLCTVS),

^{a closed subset K} of the total space X is

locally homotopy negligible

^{if the} restriction of K to each fibre

03BE-1(b)

^is

locally homotopy negligible in 03BE-1(b).

All

topological

^vector spaces

(TVS)

considered are metric

locally

^convex

topological

^vector spaces

(LCTVS).

We _say that a subset A of a

topological

space X is :

(weakly) homotopy negligible

^{if the inclusion X-A ~ X is a}

(weak) homotopy equivalence;

locally homotopy negligible

^{if each}

point

^{of A has a} fundamental

system

of neighborhoods {U}

such that the inclusion U- A - U is a

homotopy equivalence; strongly homotopy negligible

^{if for any}

neighborhood

^{U of}

X,

^{the inclusion} U- A - U is a

homotopy equivalence;

^{a Z-set}

(that ^is,

a set with

property Z,

^{see Anderson}

[1 ])

^{if for each}

homotopically

^trivial

non-empty

open subset U

of X,

^{U- A is}

non-empty

^and

homotopically

trivial.

We _say X is a manifold modeled on a space C if X has an open ^cover-

ing by

^sets

homeomorphic

to open subsets of C. For our purpose, ^a

locally

^{trivial bundle has at least the}

following

^structure:

(1)

^a

map 03BE : X ~ B,

where X is called the total space, ^B the base _space

and 03BE : X ~ B

^the

projection,

(2)

^{a space F called the}

^fibre,

(3)

for each b E

B,

^{there is an} open ^set U of b and a

homeomorphism

ç

of 03BE-1(U)

^{onto U x F such that the}

following diagram

^commutes

03BE-1(b)

is called the fibre over b.

A map

f :

X x Y - X x Z is

X-preserving

^if

f(x, y) = (x, z)

^{for each}

(x,y)EXxY.

Absolute retract

(AR),

^absolute

neighborhood

^retract

(ANR)

^are

metric _spaces in the sense of Palais

[3].

PROPOSITION 1. Let X be a

Hausdorff

paracompact

manifold

^modeled

on a closed convex subset

of

^a metrizable

locally

^convex

topological

^vector

space

(LCTVS)

^{E. For a closed subset A}

of X,

^{we have the}

following implication.

(1)

^{A is}

strongly homotopy negligible

(2)

^{A is}

locally homotopy negligible

^~

(5)

^{A is a Z-set}

(3)

^{A is}

homotopy negligible

(4)

^{A is}

weakly homotopy negligible.

PROOF.

(1)

^~

(2), (1)

^~

(5), (5)

^~

(2), (3)

^~

(4), (1)

^~

(3)

^are

trivial. It follows from the

hypothesis

^{that X is a} metrizable space which is

locally

^an

ANR,

^hence

by [3-Thm. 5],

^{X is an ANR. For an}

ANR,

(4)

^~

(3)

^{is a} well-known theorem of J. H. C. Whitehead and

(2)

^~

(1)

is the content of the main lemma of Eells and

Kuiper [2].

For

application

^of

homotopy negligible

^{subsets to various manifolds}

and the relation of it with

negligible subsets,

^{we refer to}

[3]

^and

[4].

2. Statement of the Theorems

Throughout

^this

section, by

^a

manifold

^{we shall mean a}

paracompact

Hausdorff _space modeled on

C,

^{where C is a closed convex subset of a} metric

locally

^convex

topological

^vector space

(LCTVS)

^{E. Let R denote}

the reals. The

following examples

^{of C are of interest.}

(1)

^{C = a closed convex subset of E}

(2) C = E

(3)

^{C = a}

^compact

^{convex subset of E}

(4) C = [-1, 1]n for n = 1, ···, ~

(5)

^{C =}

(- ^1, 1)n

^{for n =}

^1,...,

^oo.

121

We remark that C is a metrizable AR and X is a metrizable ANR.

The

proofs

^{of the}

^following

^theorems

(except

^{for Theorem}

3)

^{will be}

given

in section 4.

THEOREM 1. Let X be a

manifold

^{and R’ a} closed interval in R. Let K be

a closed subset of R’ X such that K ~ t X is locally homotopy negligible

in each t x

X,

^{then K is}

strongly homotopy negligible

^{in R’ x X.}

Furthermore,

^{the same is true when}

’locall y homotopy negligible’

^is

replaced by

^{’a Z-set’.}

COROLLARY 1. Let X be a Fréchet

manifold

^{and K a closed set in}

(0, 1)

^{x X} such that K n t x X is a Z-set in each t x X. Then K is a Z-set in

(0, 1)

^{x X.}

(See

Theorem 4 for a

stronger result.)

Thus we settle a

question

^{raised in}

[5-8].

THEOREM 2. Let

Xl , X2

^be

manifolds

^{and K a} closed subset

of Xl

^x

X2

such that K n x x

X2

^is

locally homotopy negligible

^{in x x}

X2 for

^each

x E

Xl.

^{Then K is}

strongly homotopy negligible

ⁱⁿ

Xl

^x

X2.

COROLLARY 2. Let X be a

manifold

^{and K a closed subset}

of

^{s x}

^X,

where s ⁼

( -1, 1)~,

^{such that each x x X n K is a Z-set in x x X. Then}

K is a Z-set in s x X.

The

following

theorem is a consequence of Theorem 2.

THEOREM 3.

Let 03BE :

^{X ~ B be a}

locally-trivial

^{bundle with}

fibre

^{F such}

that X is paracompact

Hausdorff

^{and both F and B are}

manifolds.

^{Then a}

closed subset K

of

the total space ^{X is}

strongly homotopy negligible

^{in X}

if

^K

~ 03BE-1 (b)

^is

locally homotopy negligible

^{in each}

fibre 03BE-1(b)

^{over b.}

PROOF. It suffices to observe that X is

locally

^a

product

^{of manifolds}

Ml

^x

M2,

^where

Ml, M2

^are open subsets of

B,

^F

respectively,

^{such that}

K n

(x

^x

M2)

^is

locally homotopy negligible

^{in x x}

M2

^{for each x E}

Ml.

By

^Theorem

2,

^{K n}

Ml

^x

M2

^is

strongly homotopy negligible

ⁱⁿ

Ml

^x

M2.

Then we observe that X is an ANR

[3-Thm. 5] and {M1 M2}

^{form a}

neighborhood system

^{of X. Now}

apply proposition

^1.

THEOREM 4. Let In =

[-1, 1]n

^{be the}

n-cube,

^{n =}

1, 2, ···, ~,

^and let Sn =

(-1, 1)n

^{be the interior}

(or pseudo-interior

^{in case}

of

^{n =}

~) of

I". Let X be a

manifold.

^{Then a closed subset K}

of

^{In X X is}

strongly homotopy negligible

^{in In x X}

if

^{K n}

(x

^x

X)

^is

locally homotopy negligible

in x x

X for

^{each x E sn.}

Furthermore,

^{the same is true when we}

replace ’locally homotopy negli- gible’ by

^{’a Z-set’.}

THEOREM 5. Let

Cl , C2

^be spaces such that

Cl

^{is a closed convex subset}

of

^{a metric LCTVS}

Ei. Suppose

^{K is a} closed subset

of Cl

^x

C2

^{such that}

K n

(x

^x

C2 )

^is

weakly homotopy negligible

^{in x x}

C2 for

^{each x E}

Cl ,

then K is

homotopy negligible

ⁱⁿ

Cl

^x

C2.

COROLLARY 3. Let X be a Fréchet _space or the Hilbert

cube,

^{then K is}

homotopy negligible

^{in R x X}

if

^{K n t x X is}

ewakly ^homotopy negligible

in each t x X.

THEOREM 6. Let In =

[0, 1]n

^{and sn =}

(0, 1)n,

^{n =}

1, 2, ···, ~.

^{Let C}

be any closed ^convex subset

of

^{a TVS E.}

Suppose

^{K is a closed subset}

of

In x C such

that for

^{each x E}

sn,

^{K n}

(x

^x

C)

^is

weakly homotopy negligible

in x x C. Then K is

homotopy negligible

^{in In X C.}

COROLLARY 4. Let X ⁼ a Fréchet _space or the Hilbert cube. Then a

closed set K in

[0,

¹

] x

^{X is}

homotopy negligible

ⁱⁿ

[0, 1]

^{x X}

if K

ⁿ

(t

^x

X)

is

weakly homotopy negligible

^{in each t x}

X for

^{0 t 1.}

Added in

Proof.

^Since this paper ^was

written,

^{the author in}

[6-Cor 5]

has

generalized

^{Theorem 3 to include all ANRS.}

Precisely,

^{it states that}

Theorem 3 is true when we

replace

^X

by

any metric

ANR,

^F

by

any metric space which is

locally

^{an AR}

(or manifold)

^{and B}

by

any Hausdorff space. This result is based ^{on a}

Lifting

^Theorem

[6-Thm 8].

3. Lemmas

LEMMA 1. Let

Ml, M2

^{be metric} spaces and Y an absolute

neighborhood

retract

(ANR).

^{Let K be a} closed subset

of Ml

^x

M2 . Suppose f :

^{K ~}

Ml

^{x Y is a}

M1-preserving

map,

then f

^{can be extended to a}

Ml preserving

map

of a neighborhood

^U

of K

^into

Mi

^{x Y.}

PROOF. Let P :

Ml

^{x Y ~}

Y be the projection. Then Pf :

^{K ~ Y extends}

to a map ^g of a

neighborhood

^{U of K into Y. Define F : U ~}

Ml

^{x Y}

by F(xl , x2) = (x1 , g(Xl’ X2)).

^{F is a} desired extension.

Let E n denote the Euclidean n-space

R1 R2

X - - - x

Rn

^where

Ri

^is

the real R. Let

Dn, Sn-1

^denote

respectively

the n-ball and

(n-1)-sphere

of E n. We

regard

^{En as} the subset En 0 in

En+1.

Let a b c and consider

[a, c ] as

^{a closed interval in}

Rn

+ 2.

LEMMA 2. Let X be an ANR and K a closed subset

of Rn+2

^{x X. For}

n ~

0,

^{and i =}

1, 2,

suppose

where

I1 = [b, c], I2 = [a, b],

^are

Rn+2-preserving

maps such that

(1 ) f1|b Sn-1 = f2|b Sn-1

^and

(2) f1|b Dn is homotopic to f2|b Dn in

b X - K modulo

b Sn-1, that is, the

^entire

homotopy

agrees

with f1

123

on

b Sn-1.

Then there is an

Rn+2-preserving

map F

of [a, c ] x Dn

^into

Rn+2 X-K such

^that

F|I1 Sn-1 ~ c Dn = f1

^and

F|I2 Sn-1 ~ a Dn = f2.

PROOF. We note that for n ⁼

0, Sn-1 = 0.

Thus

requirements

^on

^Sn-1

become irrelevant and

meaningless.

^{With that} modification in mind the

following proof

goes

through

^{even for this case.}

We may ^{assume a}

= -1, b = 0

^{and c = 1. Recall that}

I1 = [0, 1 ], 12 = [ -1, 0 ]

^{and we}

regard [ -1, 1 ]

^{as a} subset of

Rn+2.

^{Define for}

i ⁼

1, 2, Rn

⁺

2-preserving

maps

by

and

It is clear that

Next define a map

ho

^{of 0 x Sn into 0 x X - K}

by

By (1)

^{of the}

hypothesis fl , f2

agree ^on 0 ^x

sn-le

Hence

ho

^is well-defined.

By

^condition

(2)

^{of the}

hypothesis, ho

^{can be extended to a} map h of

Dn+1

into 0 x X-K. Let

Define a

Rn + 2-preserving

^map

Since

Rn+2 X-K

^{is an}

ANR, by

^Lemma

1,

^{0 can be extended to a}

Rn+2-preserving map 0i

^{of a}

neighborhood

^{U of A in}

En+2

into

Rn+2 X-K.

It is

elementary

^{to know} that there is a

Rn

⁺

2-preserving map H

^of

^{En + 2}

onto itself such that

(1)

^{H is the}

identity

outside of

U, (2)

^{H is one-to-one}

on

En+2-Dn+1, (3)

^H

collapses ^Dn+1

^{onto Dn}

^by H(x1, ···, xn+1)

⁼

(x1, ···, xn, 0) and (4) H|[-1,1] Sn-1 ~ {-1,1} Dn = ^identity.

We now

define

a

Rn

⁺

2-preserving

map ^{F :}

[-1, 1] Dn ~ Rn+2 X-K by

F is the desired function.

LEMMA 3. Let E be a metric TVS and u an

n-simplex

^{in R x E. Let Y be}

a metric absolute retract

(AR)

^{and K a} closed subset

of

^{R x Y such that}

K n t x Y is

weakly homotopy negligible

^{in t x Y}

for

^{each t. Then any}

R-preserving

^map

of Bd( (J)

^{into R x Y- K can be extended to an}

^R-preserv-

ing map of

^{03C3 into R x Y-K.}

PROOF. It is known that a metric AR is contractible. It follows that t x Y- K is

homotopically

trivial for each t. Let à

= boundary

(J,

03C3(t) =

J n t ^x E and

6(t)

⁼

boundary 03C3(t).

^{Let P : R x E - R be the}

projection

and let

[a, b]

⁼

P(03C3), a ~

^{b. If a =}

^b,

the lemma follows

immediately

from the

hypothesis.

^So suppose a b.

By convexity

^each

6(t)

^{is a}

(n -1 )-simplex

^{for a t b.}

Let g : à

^{~ R x Y-K be a}

R-preserving

map. At this

point

^{we divide}

the

argument

^{into two cases.}

CASE 1. n &#x3E; 1.

By hypothesis

^of

K,

^{we have for}

each t,

^a map gt :

u(t)

^{~ t x Y-K such that}

gt|03C3(t)

^{= g.} For fixed t let A = 03C3 u

6(t)

and define a

R-preserving

map

Gt :

^{A - t x}

Y- K by Gt(x)

⁼

g(x)

^when

x ~ 03C3 and

Gt(x)

⁼

gt(x)

^{when x ~}

6(t).

^{Since R x Y-K is an}

^ANR, by

Lemma

1, Gt

^can be extended to a

R-preserving map Gt of

^a

neighbor-

hood

Ut

^{of A} into R x Y-K. Since

6(t)

^is

^compact,

^{there is an open}

interval

(at, bt) containing

^{t such that}

G’(u(s»

ⁿ

^{K = Ø}

^{for all}

at ~ s ~ bt.

^Since

[a, b]

^is

compact,

finite number of

(at, bt)

^covers

[a, b].

^It follows that there are a finite number of reals a ⁼ ao a,

··· am+1 = b and a collection of

R-preserving

^maps

{fi}mi=0

^such

that for all

i,

where Ai = U{03C3(t)|ai ~ t ~ ai+1}, i = 0, 1, ···, m, and

Since

g(6(a) u 03C3(b))

^{~ K =}

^0,

^{we may assume}

f0|03C3(a1) = f1|03C3(a1)

^and

fm-1|03C3(am) = fmlu(am).

Consider fl and f2 .

It is clear that

A1, A2

^{can be}

regarded respectively

as

[a1, a2] Dn-1, [a2, a3] Dn-1.

^{At the level}

a2xDn-l

^{we have two}

125

maps f1, f2 : a2 Dn-1 ~

^a2

Y-K such that f1|a2 Sn-2 = f2|a2 Sn-2

^{= g.}

Since

a2 Y-K

^is

homotopically trivial,

^{it follows}

that f1|a2 Dn-1 is homotopic to f2|a2 Dn-1 in a2xY-K modulo a2 Sn-2. f1, f2 satisfy

all the conditions of Lemma 2. Hence there is an

R-preserving

map

Fi

^of

Al u A2

^{into R x Y-K such that}

F1|03C3

= g and

Fi =/1

^on

6(al ).

^Let

f’0 = f0, f’1 = F1|A1 and f’2 = F11A2. Thus f’0|03C3(a1) = f’1|03C3(a1) and f’1|03C3(a2) = f’2|03C3(a2). ^Repeat

the above process

for/2 and 13 .

^{It follows} that there are

R-preserving maps f"2 and f’3

^defined

respectively

^on

A2

^and

A3

^{such that}

both _agree

with g

^on

03C3, f"2|03C3(a2) = f’1|03C3(a2), f"2|03C3(a3) = f’3|03C3(a3)

^{and so on.}

Define

Since

{Ai}

^is

^finite,

^{it is clear that we can extend G to the entire}

03C3 = A0 ~ A1 ~ ··· ~ Am

^{with the}

required properties.

CASE 2. n ⁼ 1. For each

(t, y)

^{E R x}

^Y-K,

^{there is an} open interval

L, containing t

^{such that}

L,

x y ^{~ K =}

0.

Since the inclusion t x Y-K ~ t x Y is ^a

homotopy equivalence,

^{K cannot contain t x Y. It follows that}

the set of all

Lt

^covers

[a, b]. By compactness,

^{there are a} finite number

a ⁼ a0 a1 ··· am+1 = b and a finite subset

{u0, u1, ···, um}

^of

Y such that for

all i, [ai, ai+1] ui nK= 0, (ao, uo)

⁼

g(03C3(a))

^and

(am+1, um)

⁼

9 ( u (b ) ).

^Let

Ai

⁼

U {03C3(t)|ai ~ t ~ ai+1}, i

⁼

^{0, ···, m}

and

define fi : Ai ~

^{R x Y-K}

by fi(03C3(t)) = (t, u;).

^{We may assume}

fola(a1)

⁼

f1|03C3(a1) and fm-1|03C3(am)

⁼

fm|03C3(am).

^{The rest of the}

^proof

^{is the}

same as Case 1.

We are now

ready

^{to state our main Lemma.}

LEMMA 4. Let E be a metric TVS and Y a metric AR. Let K be a closed subset

of

^{R x Y} such that each K n t x Y is

weakly homotopy negligible

ⁱⁿ

t x Y. Let

ITI

^{be a}

finite simplicial complex

^{in R x E and S a}

subcomplex of T.

^{Then each}

R-preserving

map

of |S|

^{into R x Y-K can be extended}

to an

R-preserving

map

of 1 Tl

^{into R x Y- K.}

Furthermore,

^{the same is}

true when ’R’ is

replaced by

’an interval

of

^R’.

PROOF. Denote the i-skeleton of T

by Ti (that ^is,

^all

simplices

^{of T of}

dimension ~ i). Let f o

^{be an}

R-preserving of 19 ^into

^{R x Y- .K. Since}

t x Y is not contained in K for any t, ^we may

assume fo

^{is an}

R-preserving

map

of |S| ~ |T0|

^{into R Y-K.}

By

^Lemma

3, fo

^{can be extended to}

an

R-preserving map fl of |S| u |T1| into

^{R x Y-K.}

By

^{Lemma 3}

again

we can

extend fl

^{to an}

R-preserving map f2 of 19 u T2

^{into R x Y- K.}

Inductively, fo

^{extends to an}

R-preserving

map

of 1 Tl

^{into R x Y-K.}

4. Proofs

PROOF ^oF THEOREM 1. We need

only

^{to show}

(see Proposition 1)

^that

K is

locally homotopy negligible

^{in R’ x X. Let U b; a closed}

neighbor-

hood of x E X such that there is a

homeomorphism ~

^{of U onto a closed}

neighborhood

^V of C. Let J be a closed interval of R’. It follows that J x U is contractible and we want to show

03C0k(J U-K)

^{= 0 for all}

k ~

^{1. So}

suppose f : (Dn, Sn-1) ~ (J U, J U-K)

^{is a map.}

^{Let 03C8}

be the

identity

^{map of J onto J and let =}

(03C8, ~) :

^{J x U - J x V. Let}

f0 = 03BB·f

^and

Ko = 03BB(K ~ J U). Ko

^is closed in R V and

f0 :

(Dn, Sn-1) ~ (J V, J V-K0).

^{J V is a convex} subset of R x E.

By

^standard

argument

^{it follows}

that fo

^is

homotopic

^{to a}

simplical

map g of

(Dn, Sn-1)

^into

^{(J x V, J} V-K0) ^by maps ffl,

^t such that each

ft : Dn ~ J V, f1 = g and ft(Sn-1)

ⁿ

^{K0 = 0}

^{for all t.}

By hypothesis

of K and

Proposition ^1,

it follows that

K0

^{n t x V is}

homotopy negligible

in t V for each x ^E R. Since V is an

AR, by

^{Lemma 4 there is a} map y of

g(Dn)

^{into Jx}

^V-Ko

^{such that}

03BC|g(Sn-1).

⁼

^identity.

^{Let h = ,ug.}

Combining

^{with the}

homotopy {ft}

^{we therefore have} shown that there is a map

Fo :

^{nn -+ J x}

h- Ko

^{such that}

Fo Is. -, = fo.

^{Let F =}

03BB-1F0.

Thus F : D n --+ J x U- K and

F|Sn-1 = f.

^{This shows J x U- K is}

weakly homotopy negligible. ^By locally convexity of E, {J U}

^{form a}

^neighbor-

hood

system

of R’ x X. The theorem now follows from

Proposition

^1.

PROOF oF THEOREM 2. Let

Ul, U2

^{be closed}

neighborhood

^of x1 ^~

X1,

x2 ^E

X2 respectively

such that for i ⁼

1, 2,

^{there are}

homeomorphisms

~i of

Ui

^{onto a convex closed}

neighborhood Vi

^of

Ci,

^where

Ci

^{is a closed}

convex subset of a TVS

Ei. By

^local

convexity

^of

El

^and

E2, {U1 U2}

form a

neighborhood

system ^of

Xl x X2,

^so

^by Proposition 1,

^{we need}

only

^{to show}

1Ck(Ul

^x

U2-K)

^{= 0 for}

^{all k ~}

^1.

Let f : (Dn, Sn-1)

^~

( Ul

^x

U2 , Ul

^x

U2-K) be

^{a map and}

let f0 = 03BBf,

where 2 ⁼

(~1, ~2). So fo : (Dn, Sn-1) ~ (V1 V2, V1 V2-K0)

^where

K0 = 03BB(K ^{~ U1}

^x

U2).

^{It suffices to show that there is a} map

Fo :

^{Dn ~}

Vl

^x

V2-K0

^{such that}

F0|Sn-1 = f0.

^{As in Theorem}

1,

^it is well-known that there is a

homotopy {ft}

of map of ^Dn into

V, x V2

^{such that}

ft(Sn-1)

^~

^{Ko = 0}

^{for all t and g =}

fl is simplicial.

^{Let F be the finite-}

dimensional

subspace

^of

El

^x

E2 generated by g(Dn).

^Let

^{F, = Pi} (F)

where

P1 : El

^x

E2 -+ El

^{is the}

projection. Ko

is closed in

El

^x

V2, by hypothesis

^and

Proposition 1, x x V2

ⁿ

Ko

^is

locally homotopy negligible

in

x x Tl2

^{for each}

x E El . By

Theorem 1

K0 ~ (Jl x Tl2)

^is

strongly homotopy negligible

ⁱⁿ

J1 V2

^for any

1-simplex ^J1

^of

^El.

^Since

^Fl

^is

finite-dimensional,

say of

dimension n, by induction, K0

ⁿ

(Jm

^x

Tl2)

^is

strongly homotopy negligible

ⁱⁿ

Jl

^x

V2

for any m-cube Jm in

Fl .

K1 = P1g(Dn)

^{is an}

m-simplex

^of

Fi,

^hence

homeomorphism

^{to some}

127

m-cube Jm. It follows that

K0

ⁿ

(Ki

^x

Tl2 )

^is

strongly homotopy negli- gible

ⁱⁿ

K1

^x

V2,

^{which contains}

g(Sn-1).

^Thus

^g|Sn-1

^{can be extended}

to a gl of D" into

(K1

^x

v2) - Ko . Combining

^{with the}

homotopy {ft},

we have shown that there is a map

F0

^{of Dn into}

V1 V2-K

^such

that, F0|Sn-1

⁼

f0.

^The

proof

^{of the theorem is now}

complete.

PROOF oF THEOREM 4. Let U be a canonical closed

neighborhood

^{of I"}

that

is,

^{U is a}

product

^of closed intervals. Let

U1 =

U- sn. Let V be ^a closed

neighborhood

^{of X such that there is a}

homeomorphism

^{of V}

onto a closed convex subset

Vl

^{of a TVS E. For}

simplicity

^{we assume}

V1

^{is V.} It follows from the

hypothesis

^{and from Theorem 3 that}

K ~

( Ul

^x

V)

^is

homotopy negligible in Ul

^{x V. Now}

let f : (Dn, Sn-1)

~

(U x V,

^{U x}

V-K)

^{be a} map. Since

f(Sn-1) ^{~ K = Ø,}

let 03B5 &#x3E; 0 denote the distance

between f(Sn-1)

^{and K}

(distance

^{in the sense of the}

usual induced metric on the

product

^{space I n x}

X).

It is evident that there is a

homotopy {gt}

of maps of Un Sn into U such that _{go =}

identity, gl(U n sn)

^c

Ul

^and

d(gt, go)

^{8 for all t. Now define a}

homotopy

{ft}

of maps of ^{U x V} into itself

by ft(x, y) = (gt(x), y).

^{It follows that}

fo = identity, fl ( U x V)

^c

^Ul

^{x V}

and ft(Sn-1) ^{~ K = Ø}

^{for all t. Let}

ht = ft f.

^Then

h 1 : (Dn, Sn-1) ~ ( Ul ^{V, U1} V-K).

^Thus

hl fsn-l

^ex- tends to a map h of Dn into

Ul

^{x V- K.}

Combining

^{with the}

homotopy {ht},

^we have shown

that f|Sn-1

^{extends to a} map of Dn into ^{U x} V-K.

Thus U x V-K is

homotopically

trivial and the

proof

of the theorem is

complete.

PROOF oF THEOREM 5.

By

^the

Corollary

^of

[3-Thm. 15] it

^{suflices to}

show that

Ci

^x

C2-K is homotopically

^trivial.

We first consider the

special

^{case that}

Cl

^{is a closed convex subset of}

the reals R. Let

f : (Dn, Sn-1) ~ (Cl

^x

C2, Cl

^x

C2 - K)

^{be a} map. As illustrated in the

proofs

^{of the}

previous

^theorems

(which

^{are rather}

elementary)

^{we may assume}

^f

^is

simplicial. ^By

^{Lemma 4 there is a}

C1-preserving

map g

of f(Dn)

^into

el x C2 - K

^{such that}

g 1 f(s. - i)

⁼

identity.

^Hence

Cl x C2 -K

^is

homotopically

trivial and the

proof

^of

the

special

^{case is}

complete.

Now consider the

general

^{case. The}

proof

is very similar ^to that of Theorem 2. We

proceed

^{as follows.}

Let f : (Dn, Sn-1) ~ (C1 C2,

C1

^x

C2-K)

^{be a map.}

Again

^{we may}

assume f is simplicial.

^{Let F be}

the finite dimensional

subspace

^of

El

^x

E2 generated by f(Dn).

^Let

F1 = P1(F)

^where

P1 : E1 E2 ~ E1

^{is the}

projection. By

^the

special

case proven

above, K

ⁿ

(Jl

^x

C2)

^is

homotopy negligible

ⁱⁿ

Jl

^x

C2

^for

each

1-simplex Jl of El. Inductively,

^{K ~}

(Jn C2)

^is

homotopy negli- gible

ⁱⁿ any n-cube of

Fi.

^Since

K1 = P1(f(Dn))

^{is an}

m-simplex,

^which

is

homeomorphic

^{to an}

^m-cube,

^{it follows that n}

(Ki

^x

C2)

^{is homo-}

topy negligible

ⁱⁿ

K1 C2,

^which

contains f(Sn-1).

^{Hence we may make}

the extension

of f|Sn-1

^{to Dn in}

K1

^x

C2-K

^{and the}

proof

^is

complete.

PROOF oF THEOREM 6. The

proof

^{makes use} of Theorem 5 and is

exactly

^{the same as} that of Theorem 4.

REFERENCES

R. D. ANDERSON

[1] On topological ^infinite deficiency, Mich. Math. J. 14 (1967), ^365-383.

J. EELLS, Jr. and N. H. KUIPER

[2] Homotopy negligible subsets, Compositio ^Math. 21 (1969) ^155-161.

[3] R. S. PALAIS

Homotopy theory of infinite dimensional manifolds, Topology ⁵ (1966) ^1-16.

R. D. ANDERSON, ^D. HENDERSON, ^{and J. WEST}

[4] Negligible subsets of infinite-dimensional manifolds, Compositio ^{Math. 21} (1969)

143-150.

[5] Problems in Infinite-dimensional spaces and manifolds, Manuscript, LSU, ^Baton Rouge, La. 1969.

[6] ^On homeomorphisms ^of infinite-dimensional bundles, I, ^to appear.

(Oblatum 18-111-71) Department of Mathematics University of California Santa Barbara, Calif. 93106 U.S.A.